What the test question that just bugged the crap outta me

1. Sep 28, 2005

schattenjaeger

we were given the DE y'=2sqrt(y) that was how it was given, I didn't take the square root myself and thus forget a +and-

so using a bernoulli equation I got (x+c)^2(I think, I'm doing this from memory)

then the next part asked to find two solution curves for the point (1,1), so x^2 and (x-2)^2, right?

THEN it asked for the solution guaranteed by the existance and uniqueness theorem at (1,1), but didn't I just find TWO solution curves for that same point, which means there ISN'T an unique solution? And to further confuzzle matters if I actually apply the theorem it seems like it SHOULD have an unique solution at (1,1)

2. Sep 28, 2005

Tom Mattson

Staff Emeritus
Take a closer look at the solution $y=(x-2)^2$. It has a negative slope at $x=1$. If $y'<0$, then according to your DE $2\sqrt{y}<0$, which is impossible.

3. Sep 29, 2005

schattenjaeger

aw hell, so when it asked me to sketch two solution curves THAT was the "trick" question?

4. Sep 30, 2005

schattenjaeger

or were they both solution curves but they passed through the point or something? We'll go over it on Monday but I hate waiting to know

5. Sep 30, 2005

Tom Mattson

Staff Emeritus
They are not both solutions at $x=1$. $y=(x-2)^2$ simply does not satisfy the differential equation at $x=1$. What happened was that you introduced an extraneous root when you squared $y^{1/2}$.